Could we extend isomorphisms between cohomologies of h-injective complexes to h-injective complexes themselves?

Let $ R$ be an associative ring with unit and $ I$ be a complex of $ R$ -modules. We call $ I$ is h-injective if for any acyclic complex $ T$ of $ R$ -modules, the mapping complex $ \text{Hom}_R(T,I)$ is acyclic too.

Now let $ I$ and $ J$ two h-injective complexes of $ R$ -modules. Let $ H^{\cdot}(I)$ and $ H^{\cdot}(J)$ be the cohomologies of $ I$ and $ J$ .

My question is: suppose we have maps of $ R$ -modules $ f^n:H^n(I)\overset{\sim}{\to}H^n(J)$ for each $ n$ , could we define a quasi-isomorphism $ $ \phi: I\overset{\sim}{\to} J $ $ such that $ H^n(\phi)=f^n$ for each $ n$ ? We may add some condition such as both $ I$ and $ J$ are componentwisely injective and have bounded cohomologies.